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My obsessive nature is finally satisfied with this problem. After proving that Poisson Distribution becomes symmetric as the expected value, lambda, approached infinity, something never sat right with me. It came to me that I never proved the probability of lambda occurrences occurring as lambda goes to infinity was zero. The graphs above show that as lambda increases, the probability decreases. I assumed it goes to zero. I can think of a couple hand-waves that state it can't be non-zero, but nothing analytically grounded. I started attacking that limit only to be stumped by different ways to simplify the limit without gaining any ground. I found the Stirling Approximation, which approximates large factorials, solved it easily.
It was quite serendipitous how this happened, too. I was talking with my intro to astronomy/astrophysics professor, and he had what seemed to be a work book of math problems. It was actually a table that helped calculation something or other and he was investigating how good the approximations of the chart were when using large numbers. He simultaneously had an actual math book out that explained the Stirling Approximation to help him with the calculations. Curious about this Stirling Approximation, he explained it was used to approximated large factorials. At the time, this limit hadn't crossed my mind. When checking the Approximation on Wikipedia, I immediately recognized the terms to be part of one of my simplifications.
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